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Rational Series in the Free Group and the Connes Operator

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Rational Series in the Free Group and the Connes Operator RATIONAL SERIES IN THE FREE GROUP AND THE CONNES OPERATOR AARON LAUVE AND CHRISTOPHE REUTENAUER Abstract. We characterize rational series over the free group by using an oper- ator introduced by A. Connes. We prove that rational Malcev–Neumann series posses rational expressions without simplifications. Finally, we develop an effec- tive algorithm for solving the word problem in the free skew field. 1. Introduction Fix a field k and a finite set of indeterminants X. The free commutative field k(X) is defined as the initial object in the category of (commutative) fields F with embeddings k[X]֒→F . It also has a well understood realization as the field of rational functions: k(X)= f/g | f, g ∈ k[X], g = 0 . The focus of this paper is the noncommutative counterpart to k(X), the free (skew) field k<(X>.) It has an analogous categorical definition involving embeddings of kX, but one can hardly say that k<(X>) is well understood. As an illustration, the reader may take a moment to verify that x − z−1 1 − yx −1 y − z + y−1 − z−1 1 − x−1y−1 −1 x−1 − z = 0 (1.1) without allowing the variables to commute. A commonly used realization of the free field, due to Lewin [Lew74], involves Malcev–Neumann series over the free group Γ(X) generated by X. These are noncommutative, multivariate analogs of Laurent series; we recall the details in Section 2. While the Lewin realization of the free field is powerful, it suffers from an incon- venient asymmetry. We illustrate with an identity of Euler [Car00]: xk = 0. Z k∈ −k k This looks absurd, but rewriting it as k≥0 x + x k≥0 x , we recognize the geometric series expansions of 1/(1 − x−1) and x/(1 − x), respectively. Adding these fractions together does indeed give zero. Of course, kJX ∪ X−1K is not a ring, so the mixing of series in x and x−1 is usually disallowed. Nevertheless, passing through steps such as this one is a useful technique for proving identities in the free field. (See Appendix A.) Date: July 4, 2012. 1 2 AARONLAUVEANDCHRISTOPHEREUTENAUER In this paper, we give Euler’s identity firm footing in the noncommutative setting. (See [BHS09, BR03] for more on the commutative setting.) Our main result in this vein is a characterization of which series over the free group Γ(X) have rational expressions (Theorem 3.1 of Section 3). The main ingredient in the proof is a Fredholm operator F on the free group that we describe in Section 2.3. This operator featured prominently in a conjecture of Alain Connes [Con94] that was proven in [DR97]. Our result extends the main result there. Briefly, an element a ∈ kΓ is rational if and only if its associated Connes operator [F, a] has finite rank. In this paper, we also tackle the problem of rendering a given rational expression into Malcev–Neumann form (Section 4). Our main result here, Theorem 4.9, is an effective algorithm for solving the word problem in k<(X>) . Recall that the word problem for free groups (when are two words equal in Γ(X)?) is decidable, even though this is famously not the case for all groups [Nov55, Boo58]. The analogous problem for the free field (when are two expressions equal in k<(X>) ?) was first considered in [Coh73] and taken up again in [CR99]. Here is a simple example: Are x−1 (1 − x)−1 and x−1 + (1 − x)−1 equal? The answer is, “yes,” as the reader may easily verify. While algorithms are presented in [Coh73, CR99], their complexity seems very high. We derive our algorithm from Fliess’ proof [Fli71] of the following fact: simplifications in rational expressions preserve rationality. Here, at last, we may say that the word problem is certainly not undecidable. Though the complexity of our algorithm could certainly be improved. 2. The Free Field, Rational Series, and the Connes Operator 2.1. The free field. The free field k<(X>) is defined as the initial object in the category of epic kX skew fields with specializations. (See [Coh95, Ch. 4], which also contains a realization in terms of full matrices over kX.) In what follows, we use Lewin’s realization in terms of formal series. 2.1.1. Operations on series over the free group. Let Γ = Γ(X) denote the free group generated by a finite set of noncommuting indeterminants X. Let kΓ denote the vector space with basis Γ and kΓ denote the formal series over Γ, i.e., functions a: Γ → k, which we may write as ω∈Γ(a, ω) ω or ω∈Γ aω ω as convenient. The support supp(a) of a series a is the set of its nonzero coefficients, {ω ∈ Γ | (a, ω) = 0}. The sum a + b of two series over Γ is well-defined and defined by (a + b, ω) = (a, ω) + (b, ω) for all ω ∈ Γ. The Cauchy product a · b (or ab) of two series is well-defined if (ab, ω) := (a, α)(b, β) α,β∈Γ αβ=ω is a finite sum for all ω ∈ Γ. We will also have occasion to use the Hadamard product a ⊙ b, defined by (a ⊙ b, ω) := (a, ω)(b, ω) for all ω ∈ Γ. We define now a unary operation called the star operation: given a ∈ kΓ, we let a∗ denote the sum 1+ a + a2 + · · · , if this is a well-defined element of kΓ; in other words, if for any ω ∈ Γ, there are only finitely many tuples (ω1,...,ωn) such that RATIONALSERIESANDTHECONNESOPERATOR 3 Γ ω = ω1 · · · ωn and (a, ω1) · · · (a, ωn) = 0. In this case, it is the inverse of 1 − a in k under the Cauchy product. 2.1.2. Rational series over the free group. Fix a total order < on Γ compatible with its group structure. (See Appendix C for an elementary example.) The Malcev– Neumann series (with respect to <) is the subset k((Γ)) ⊆ kΓ of series with well- ordered support. That is, a ∈ k((Γ)) if every nonempty set A ⊆ supp(a) has a minimum element. One shows that k((Γ)) is closed under addition, multiplication and taking inverses, i.e., k((Γ)) is a skew field. See [Coh95, Ch. 2.4], [Pas85, Ch. 13.2], or [Sak09, Ch. IV.4]. The inverse of a Malcev–Neumann series a is built in a geometric series–type manner: if a = aoωo + ωo<ω aωω, then −1 ∗ −1 −1 −1 −1 −1 a = 1 − aω ωωo aoωo = ao ωo aω ωωo , (2.1) ωo<ω ωo<ω −1 −1 −1 with aω = −aωao and ωo being the minimum element of supp (a ). Lewin [Lew74] showed that the free field k<(X>) is isomorphic to the rational closure of kX in k((Γ)), i.e., the smallest subring of k((Γ)) containing kX and closed under addition, multiplication and taking inverses. An alternative proof, using Cohn’s realization of the free field in terms of full matrices, appears in [Reu99]. 2.2. Rational series over free monoids. The rational series in k((Γ)) introduced above are a particular case of rational series over an alphabet A. In this more general notion, one starts with polynomials on the free monoid kA∗ and builds expressions with (+, ×, (·)∗). (Here, for an expression p ∈ kA∗ without constant term, p∗ := 1+ p + p2 + · · · .) Indeed, let X = {x | x ∈ X} represent formal inverses for the elements of X, and put A = X ∪ X. Then (2.1) allows us to replace the (·)−1 operation over Γ with the (·)∗ operation over A. We do so freely in what follows and exploit results from the latter theory that we collect here. ∗ 2.2.1. Hankel rank. Let S = w(S, w)w be a series over A . Given a nonempty word u ∈ A∗, we define the new series u ◦ S ∈ kA, a (right) translate of S, by ◦ u S := (S, w)wo. w : w=wou Letting ε denote the empty word, put ε ◦ S := (S, ε) and extend bi-linearly to view ◦ as a map ◦ : kA∗ ⊗ kA → kA. (See [Sak09, Ch. III.4] or [BR11, Ch. 1.5], where the image is denoted Su−1.) The Hankel rank of the series S, introduced in [Fli74], is the rank of the operator ◦ S : kA∗ → kA. We extend this construction to series over the free group Γ in Section 3.4. The following is classical. Theorem 2.1 (Fliess). A series S ∈ kA over a free monoid A∗ is rational if and only if its right translates {f ◦ S | f ∈ kA∗} span a finite dimensional subspace of kA. An easy corollary is that the Hadamard product preserves rationality. For a proof, see [BR11, Th. 1.5.5]. 4 AARONLAUVEANDCHRISTOPHEREUTENAUER Theorem 2.2 (Sch¨utzenberger). If S, T are rational series, then S ⊙ T is rational. 2.2.2. Recognizable series. We recall some additional results on recognizable series that will be useful in Section 4. Fix a series S = w(S, w)w ∈ kA. Let L = supp(S). The language L is said to be recognizable if there is an au- tomaton1 that accepts L and no other words. For example, the machine in Figure 1 accepts supp(b + ac)∗; the input state is marked with an arrow and the output state is doubly circled. More generally, we may extend the notion of automata so that a b 0 1 c Figure 1. An automaton recognizing the language supp(b + ac)∗. edge-labels carry coefficients other than 0 or 1, e.g., taking values in some semiring k, and speak about S being recognizable by a k-automaton.
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